Material Properties

Parameters needed to define the material models can be found in Table 3.5. As seen in Table 3.5, there are multiple parts of the material model for each element.

Table 3.5 – Material Models For the Calibration Model

Material Model Number 1 refers to the Solid65 element. The Solid65 element requires linear isotropic and multilinear isotropic material properties to properly model concrete. The multilinear isotropic material uses the von Mises failure criterion along with the Willam and Warnke (1974) model to define the failure of the concrete. EX is the modulus of elasticity of the concrete (E ), and PRXY is the Poisson’s ratio (Q ). The _c modulus was based on the equation,

57000 '

c c

E f (3.1)

with a value of f equal to 4,800 psi. Poisson’s ratio was assumed to be 0.3. The _c^' compressive uniaxial stress-strain relationship for the concrete model was obtained using the following equations to compute the multilinear isotropic stress-strain curve for the concrete (MacGregor 1992)

H0= strain at the ultimate compressive strength f_c^'

The multilinear isotropic stress-strain implemented requires the first point of the curve to be defined by the user. It must satisfy Hooke’s Law;

E V

H (3.5)

The multilinear curve is used to help with convergence of the nonlinear solution algorithm.

0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035

Strain (in./in.)

Stress (psi)

Figure 3.8 – Uniaxial Stress-Strain Curve

Figure 3.8 shows the stress-strain relationship used for this study and is based on work done by Kachlakev, et al. (2001). Point 1, defined as 0.30f , is calculated in the _c^' linear range (Equation 3.4). Points 2, 3, and 4 are calculated from Equation 3.2 with H₀ obtained from Equation 3.3. Strains were selected and the stress was calculated for each

H

strain. Point 5 is defined at f and _c^' ₀ 0.003 . in .

H in indicating traditional crushing strain for unconfined concrete.

Implementation of the Willam and Warnke (1974) material model in ANSYS requires that different constants be defined. These 9 constants are: (SAS 2003)

1. Shear transfer coefficients for an open crack;

2. Shear transfer coefficients for a closed crack;

3. Uniaxial tensile cracking stress;

4. Uniaxial crushing stress (positive);

5. Biaxial crushing stress (positive);

6. Ambient hydrostatic stress state for use with constants 7 and 8;

7. Biaxial crushing stress (positive) under the ambient hydrostatic stress state (constant 6);

8. Uniaxial crushing stress (positive) under the ambient hydrostatic stress state (constant 6);

9. Stiffness multiplier for cracked tensile condition.

Typical shear transfer coefficients range from 0.0 to 1.0, with 0.0 representing a smooth crack (complete loss of shear transfer) and 1.0 representing a rough crack (no loss of shear transfer). The shear transfer coefficients for open and closed cracks were

determined using the work of Kachlakev, et al. (2001) as a basis. Convergence problems occurred when the shear transfer coefficient for the open crack dropped below 0.2. No deviation of the response occurs with the change of the coefficient. Therefore, the

coefficient for the open crack was set to 0.3 (Table 3.4). The uniaxial cracking stress was based upon the modulus of rupture. This value is determined using,

7.5 '

r c

f f (3.6)

The uniaxial crushing stress in this model was based on the uniaxial unconfined compressive strength ( f ) and is denoted as _c^' f . It was entered as -1 to turn off the _t crushing capability of the concrete element as suggested by past researchers (Kachlakev, et al. 2001). Convergence problems have been repeated when the crushing capability was turned on.

The biaxial crushing stress refers to the ultimate biaxial compressive strength ( f ). The ambient hydrostatic stress state is denoted as _cb^' V_h. This stress state is defined as:

1( )

h 3 xp yp zp

V V V V (3.7)

where V_xp^,Vyp^{, and} Vzp are the principal stresses in the principal directions. The biaxial crushing stress under the ambient hydrostatic stress state refers to the ultimate

compressive strength for a state of biaxial compression superimposed on the hydrostatic stress state ( f ). The uniaxial crushing stress under the ambient hydrostatic stress state ₁ refers to the ultimate compressive strength for a state of uniaxial compression

superimposed on the hydrostatic stress state ( f ). The failure surface can be defined ₂ with a minimum of two constants, f and _t f . The remainder of the variables in the _c^' concrete model are left to default based on these equations: (SAS 2003)

' '

These stress states are only valid for stress states satisfying the condition 3 '

h fc

V d (3.11)

Material Model Number 2 refers to the Solid45 element. The Solid45 element is being used for the steel plates at loading points and supports on the beam. Therefore, this element is modeled as a linear isotropic element with a modulus of elasticity for the steel (E ), and poisson’s ratio (0.3). _s

Material Model Number 3 refers to the Link8 element. The Link8 element is being used for all the steel reinforcement in the beam and it is assumed to be bilinear isotropic. Bilinear isotropic material is also based on the von Mises failure criteria. The bilinear model requires the yield stress ( f ), as well as the hardening modulus of the steel _y to be defined. The yield stress was defined as 60,000 psi, and the hardening modulus was 2900 psi.

Note that the density for the concrete was not added in the material model. For the control beam in Buckhouse (1997), the LVDT’s used to measure deflection at mid-span were put on the beam after it was set in the test fixture. Deflections were taken relative to a zero deflection point after the self-weight was introduced. Therefore, the self-weight was not introduced in this calibration model.

3.2.4 Modeling

The beam, plates, and supports were modeled as volumes. Since a quarter of the beam is being modeled, the model is 93 in. long, with a cross-section of 5 in. x 18 in. The

dimensions for the concrete volume are shown in Table 3.6. The zero values for the Z-coordinates coincide with the center of the cross-section for the concrete beam.

Table 3.6 – Dimensions for Concrete, Steel Plate, and Steel Support Volumes

ANSYS Concrete (in.) Steel Plate (in.) Steel Support (in.) X1,X2 X-coordinates 0 93 60 66 1.5 4.5

Y1,Y2 Y-coordinates 0 18 18 19 0 -1

Z1,Z2 Z-coordinates 0 5 0 5 0 5

The 93 in. dimension for the X-coordinates is the mid-span of the beam. Due to

symmetry, only one loading plate and one support plate are needed. The support is a 3 in.

x 5 in. x 1 in. steel plate, while the plate at the load point is 6 in. x 5 in. x 1 in. The dimensions for the plate and support are shown in Table 3.6. The combined volumes of the plate, support, and beam are shown in Figure 3.9. The FE mesh for the beam model is shown in Figure 3.10.

Figure 3.9 – Volumes Created in ANSYS

Steel Support

Steel Loading Plate Concrete Beam

Figure 3.10 – Mesh of the Concrete, Steel Plate, and Steel Support

Link8 elements were used to create the flexural and shear reinforcement.

Reinforcement exists at a plane of symmetry and in the beam. The area of steel at the plane of symmetry is one half the normal area for a #5 bar because one half of the bar is cut off. Shear stirrups are modeled throughout the beam. Only half of the stirrup is modeled because of the symmetry of the beam. Figure 3.11 illustrates that the rebar shares the same nodes at the points that it intersects the shear stirrups. The element type number, material number, and real constant set number for the calibration model were set for each mesh as shown in Table 3.7.

Steel Plate Element

Figure 3.11 – Reinforcement Configuration

Table 3.7 – Mesh Attributes for the Model

Model Parts Element

To obtain good results from the Solid65 element, the use of a rectangular mesh is recommended. Therefore, the mesh was set up such that square or rectangular elements

#3 Shear Stirrups

#5 Bar Reinforcement located 2.5 in.

from the end of the Cross-Section

Shared nodes of

were created (Figure 3.10). The volume sweep command was used to mesh the steel plate and support. This properly sets the width and length of elements in the plates to be consistent with the elements and nodes in the concrete portions of the model.

The overall mesh of the concrete, plate, and support volumes is shown in Figure 3.10. The necessary element divisions are noted. The meshing of the reinforcement is a special case compared to the volumes. No mesh of the reinforcement is needed because individual elements were created in the modeling through the nodes created by the mesh of the concrete volume. However, the necessary mesh attributes as described above need to be set before each section of the reinforcement is created.